10.6: Time Development is a Canonical Transformation Generated by the Action

The transformation from the variables \(\begin{equation}
q_{i}^{(1)}, p_{i}^{(1)} \text { at time } t_{1} \text { to } q_{i}^{(2)}, \quad p_{i}^{(2)} \text { at a later time } t_{2}
\end{equation}\) has to be canonical, since the system obeys Hamilton’s (canonical!) equations at all times.

In fact, the variation of the action along the true path from \(\begin{equation}
q_{i}^{(1)} \text { at time } t_{1} \text { to } q_{i}^{(2)} \text { at } t_{2}
\end{equation}\) with respect to final and initial coordinates and times was found earlier to be

\begin{equation}
d S\left(q_{i}^{(1)}, q_{i}^{(2)}, t_{2}, t_{1}\right)=\sum_{i} p_{i}^{(2)} d q_{i}^{(2)}-\sum_{i} p_{i}^{(1)} d q_{i}^{(1)}+H^{(2)} d t_{2}-H^{(1)} d t_{1}
\end{equation}

and, comparing that expression with the differential form of a canonical transformation corresponding to \(\begin{equation}
F(q \rightarrow Q, p \rightarrow P)
\end{equation}\) in the discussion above, which was

\begin{equation}
d F=\sum_{i} p_{i} d q_{i}-\sum_{i} P_{i} d Q_{i}+\left(H^{(1)}-H^{(2)}\right) d t
\end{equation}

we see that the action itself is the generating function for the canonical transformation from the variables \(\begin{equation}
q_{i}^{(1)}, p_{i}^{(1)} \text { at }
\end{equation}\) time \(\begin{equation}
t_{1} \text { to the set } q_{i}^{(2)}, p_{i}^{(2)} \text { at the later time } t_{2}
\end{equation}\) actually −S generates the forward motion in time, the equivalent variables in the two equations above being

\begin{equation}
p_{i}^{(1)} \equiv p_{i}, d q_{i}^{(1)} \equiv d q_{i}, p_{i}^{(2)} \equiv P_{i}, d q_{i}^{(2)} \equiv d Q_{i}
\end{equation}